부산대학교 도서관

A group $G$ is {\it amenable} if there exists a finitely additivetranslation invariant probability measure on all subsets of $G$. (Thisdefinition from 1929 is due to J. von~Neumann, and there are equivalentcombinatorial definitions in terms of the Cayley graph of $G$.) Theclass of elementary amenable groups is the smallest class of groupsthat contains all finite groups, all abelian groups, and is closedunder taking subgroups, quotients, extensions, and directed unions.Von~Neumann had observed that all elementary amenable groups areamenable, but it was some time before any other examples of amenablegroups were found. Chou proved in 1980 that elementary amenable groupshave polynomial or exponential growth, and the much studied {\it Grigorchuk group} from 1983, which acts on a rooted tree, is amenablebut has intermediate growth, so it cannot be elementary amenable.\par It was proved in [L. Bartholdi, V.~A. Kaimanovich and V.~V.Nekrashevych, Duke Math. J. {\bf 154} (2010), no.~3, 575--598; MR2730578] that groups acting on rooted trees with {\it bounded activity} are amenable; these are the so-called {\it boundedautomata groups}. K. Juschenko proved the first results concerningelementary amenable groups that act on rooted trees in [J. Topol.Anal. {\bf 10} (2018), no.~1, 35--45; MR3737508].\par The question addressed in the paper under review is: ``What doelementary amenable bounded automata groups look like? How close arethey to abelian?'' The questions are answered for three large classesof bounded automata groups, which contain many of the examples thathave been previously studied. Two of these classes are new. Inaddition, several examples are presented that place restrictions onwhat ``close to abelian'' might mean.\par These three classes are iterated monodromy groups, generalizedbasilica groups, and groups of abelian wreath type (which contain theGupta-Sidki groups). The principal results proved are:\roster\item"(i)" Every iterated monodromy group of a post-critically finite polynomialis either virtually abelian or not elementary amenable.\item"(ii)" Every generalized basilica group is either (locallyfinite)-by-(virtually abelian) or not elementary amenable.\item"(iii)" Every self-replicating group of abelian wreath type is eithervirtually abelian or not elementary amenable.\endroster\par In addition, the concept of an {\it odometer} in a group isintroduced, and it is proved that every bounded automata group thatcontains an odometer is either virtually abelian or not elementaryamenable.\par The paper is long but generally well written and reasonablyself-contained.